I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “bilinear estimates”. In addition to the ${L}^{2}$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the ${L}^{p}$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss...

I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “Bilinear Estimates”. In addition to the ${L}^{2}$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the ${L}^{p}$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss...

We review recent results concerning the study of rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. We develop new analytic methods based on Strichartz type inequalities which results in a gain of half a derivative relative to the classical result. Our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates. The latter allows us to take full advantage of the specific structure of...

We investigate the multiplicative properties of the spaces ${H}^{\frac{n}{2},\frac{1}{2}}$ As in the case of the classical Sobolev spaces ${H}^{\frac{n}{2}}$ this space does not form an algebra. We investigate instead the space ${H}^{\frac{n}{2}}\cap {L}^{\infty}$ , more precisely a subspace of it formed by products of solutions of the homogeneous wave equation with data in ${H}^{\frac{n}{2}}$.

We provide ${L}^{1}$ estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff–Sobolev parametrices
in a Lorentzian space-time satisfying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates
is of a more general interest.

We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation ${\square}_{\mathbf{g}}\phi =0$, where $\gg $ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes ${L}^{2}$ bounds on the curvature tensor $\mathbf{R}$ of $\gg $ is a major step towards the proof of the bounded ${L}^{2}$ curvature conjecture.

This paper reports on the recent proof of the bounded ${L}^{2}$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the ${L}^{2}$-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

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